Electronic Journal of Differential Equations, Vol. 2006(2006), No. 56, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
PERIODIC SOLUTIONS FOR SOME PARTIAL NEUTRAL
FUNCTIONAL DIFFERENTIAL EQUATIONS
RACHID BENKHALTI, ABDELHAI ELAZZOUZI, KHALIL EZZINBI
Abstract. In this work, we study the existence of periodic solutions for partial neutral functional differential equation. We assume that the linear part is
not necessarily densely defined and satisfies the Hille-Yosida condition. In the
nonhomogeneous linear case, we prove that the existence of a bounded solution
on R+ implies the existence of a periodic solution. In nonlinear case, we use
the concept of boundedness and ultimate boundedness to prove the existence
of periodic solutions.
1. Introduction
The aim of this work is to study the existence of a periodic solution for the
partial neutral functional differential equation
d
D(ut ) = AD(ut ) + F (t, ut ) for t ≥ 0
dt
(1.1)
u0 = ϕ, ϕ ∈ C := C([−r, 0]; X),
where A is not necessarily densely defined linear operator on a Banach space X. We
suppose that A satisfies the Hille-Yosida condition, which means that there exist
M ≥ 1, ω ∈ R such that (ω, +∞) ⊂ ρ(A) and
|R(λ, A)n | ≤
M
(λ − ω)n
for n ∈ N, λ > ω,
where ρ(A) is the resolvent set of A and R(λ, A) = (λ − A)−1 . Here C is the space
of continuous functions from [−r, 0] to X endowed with the uniform norm topology,
and D : C → X is a bounded linear operator which is given by
Z 0
Dϕ := ϕ(0) −
[dη(θ)]ϕ(θ) for ϕ ∈ C,
−r
for a mapping η : [−r, 0] → L(X) of bounded variation and non atomic at zero,
which means that
var (η) → 0
as → 0.
[−,0]
2000 Mathematics Subject Classification. 34C25, 34D40, 34K40, 34K60.
Key words and phrases. Integral solutions; Hille-Yosida condition; boundedness;
ultimate boundedness; condensing map; Hale and Lunel’s fixed point theorem.
c
2006
Texas State University - San Marcos.
Submitted November 14, 2005. Published April 28, 2006.
Research is supported by grant 03-030 RG/MATHS/AF/AC from TWAS.
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R. BENKHALTI, A. ELAZZOUZI, K. EZZINBI
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L(X) is the space of bounded linear operators from X into X. For every t ≥ 0, as
usual, the history function ut ∈ C is defined by
ut (θ) = u(t + θ)
for θ ∈ [−r, 0].
F is a continuous function from R+ × C into X which is periodic in t.
The theory of functional differential equations of neutral type has been developed
recently by several authors, for instance we refer to [1, 2, 3, 4, 6, 8, 18, 19, 25, 26, 27].
In [26] and [27], the authors studied neutral partial functional differential-difference
equation defined on the unit circle S, which is a model for a continuous circular array
of resistively coupled transmission lines with mixed initial boundary conditions
∂2
d
[u(., t) − qu(., t − r)] = k 2 [u(., t) − qu(., t − r)] + ζ(ut ) for t ≥ 0, (1.2)
dt
∂x
where x ∈ S, k is a positive constant, ζ is a continuous function and 0 ≤ q < 1.
The phase space is C([−r, 0], H 1 (S)). In [18, 19], the author studied the qualitative
behavior of solutions of equation (1.2), and obtained several results about stability,
attractiveness of solutions and bifurcation of solutions near an equilibrium. The
idea of studying partial neutral functional differential equations with operators
satisfying Hille-Yosida condition, begins with [2], where the authors studied the
following class of equation
d
[u(t) − Gu(t − r)] = A[u(t) − Gu(t − r)] + P (ut ) + Qu(t − r),
dt
where A satisfies the Hille-Yosida condition, G and Q are bounded linear operators from X into X and P is a bounded linear operator from C into X. It has
been proved in particular, that the solutions generate a locally Lipschitz continuous integrated semigroup. In [5], the authors studied the existence, uniqueness and
regularity of solutions of (1.1). They obtained several results concerning dissipativeness and existence of global attractor.
One of the most attractive areas of the qualitative theory of partial neutral
functional differential equations is the existence of periodic solutions. Naturally,
fixed point theorems play a significant role in the investigation of the existence of
periodic solutions. In finite dimensional spaces, many works are devoted to this
subject. In [11] and [16], using Horn’s fixed point theorem, the authors proved
that if the solutions of an n-dimensional periodic ordinary differential equation are
bounded and ultimately bounded, then the system has a periodic solution. In [10],
the authors gave several criteria for the existence of periodic solutions of functional
differential equations with infinite delay, they obtained the existence of periodic
solutions by using Sadovskii’s fixed point theorem. In [20] and [22] the authors
used Horn’s fixed point theorem to prove the existence of periodic solutions for
functional differential equations with finite delay. Recently, the authors in [7],
studied the following partial neutral functional differential equation
d
D(ut ) = AD(ut ) + L(ut ) + g(t) for t ≥ 0,
(1.3)
dt
where A satisfies the Hille-Yosida condition, L is a bounded linear operator from
C into X and g is a continuous function for R+ to X. They established a variation
of constants formula for equation (1.3). This formula is used to prove the existence
of bounded, periodic and almost periodic solution when the solution semigroup of
equation (1.3) with g = 0 is hyperbolic. Recall that the main approach to prove the
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PERIODIC SOLUTIONS
3
existence of periodic solutions, is to consider the Poincaré map P which is defined
by
Pϕ = uω (., ϕ),
where u(., ϕ) is the solution of (1.1). Then one establishes the existence of fixed
points of P which are the initial values of periodic solutions.
In [15, 23], the authors used the Poincaré map and they proved the existence
of periodic solutions for nonlinear partial functional differential equations of retarded type which correspond to Dϕ = ϕ(0), they used the boundedness and the
ultimate boundedness of solutions to get a periodic solution by using Horn’s fixed
point theorem which requires the compactness of the solution operator. For partial
neutral functional differential equations, the Poincaré map P is not compact, and
fixed point theorems requiring compactness couldn’t be used. We consider the case
where F is linear with respect to the second argument, we show that the existence
of a bounded solution on R+ implies the existence of a periodic solution. To achieve
this goal, we use Chow and Hale’s fixed point theorem for affine maps [12] to prove
that the Poincaré map P has at least one fixed point. For the nonlinear case, we
use the boundedness and the ultimate boundedness and we prove the existence
of periodic solutions by using Hale and Lunel’s fixed point theorem which is an
extension of Horn’s fixed point theorem for condensing maps.
The work is organized as follows: in section 2, we give some definitions and results
about the solutions of (1.1). In section 3, we discuss the existence of periodic
solutions where F is linear with respect to the second argument. In section 4,
we study the existence of periodic solutions in the nonlinear case, we assume that
solutions are bounded and ultimate bounded. Finally, we propose some applications
for some partial neutral functional differential equations with diffusion.
2. Existence and estimation of solutions
Throughout this work, we suppose that
(H0) A satisfies the Hille-Yosida condition.
The following results concern the existence of integral solutions of (1.1).
Definition 2.1 ([3, 5]). A continuous function u from [−r, T ] to X with T > 0, is
an integral solution of (1.1) if
Rt
(i) 0 D(us )ds ∈ D(A) for t ∈ [0, T ],
Rt
Rt
(ii) D(ut ) = Dϕ + A 0 D(us )ds + 0 F (s, us )ds for t ∈ [0, T ],
(iii) u0 = ϕ.
From the closedness property of A, one can see that if u is an integral solution of
(1.1), then D(ut ) ∈ D(A) for all t ∈ [0, T ]. In particular, Dϕ ∈ D(A). It has been
proved in [3], that the condition Dϕ ∈ D(A) is enough for the existence of integral
solutions of (1.1). The part A0 of the operator A in D(A) is defined by
D(A0 ) = {x ∈ D(A) : Ax ∈ D(A)},
A0 x = Ax
for x ∈ D(A0 ).
Lemma 2.2. [9] A0 generates a strongly continuous semigroup (T0 (t))t≥0 on D(A).
For the existence of the integral solutions, we assume that
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R. BENKHALTI, A. ELAZZOUZI, K. EZZINBI
EJDE-2006/56
(H1) F is continuous and Lipschitzian with respect to the second argument:
There exists a positive constant µ such that
|F (t, φ) − F (t, ψ)| ≤ µ|φ − ψ| for φ, ψ ∈ C, t ≥ 0.
Theorem 2.3 ([3, Theorem 2]). Assume that (H0) and (H1) hold. Then, for all
ϕ ∈ C such that Dϕ ∈ D(A), there exists a unique integral solution u of (1.1) on
[0, +∞). Moreover, u is given by
Z t
D(ut ) = T0 (t)Dϕ + lim
T0 (t − s)Bλ F (s, us )ds for t ≥ 0,
(2.1)
λ→+∞
0
where Bλ = λR(λ, A) for λ > ω.
In the sequel, integral solutions will be called solutions.
Proposition 2.4. Assume that (H0) and (H1) hold. Let u and v be solutions of
e such that
(1.1) on [−r, T ] for T > 0. Then, there exist positive constants N and N
|ut − vt | ≤ N eN t |u0 − v0 |
e
for t ∈ [0, T ].
(2.2)
This is an immediate consequence of the following fundamental lemma.
Lemma 2.5 ([3, Lemma 5]). There are positive constants a, b and c such that for
any continuous function h : R+ → X, the solution w of the difference equation
D(wt ) = h(t) for t ≥ 0
w0 = ϕ.
satisfies the estimate
h
i
|wt (., ϕ)| ≤ exp(at) b|w0 | + c sup |h(s)|
for t ≥ 0.
(2.3)
s∈[0,t]
Proof of Proposition 2.4. Let u and v be two solutions of (1.1) on [−r, T ], for some
T > 0. Then, for t ∈ [0, T ]
Z t
D(ut − vt ) = T0 (t)D(u0 − v0 ) + lim
T0 (t − s)Bλ (F (s, us ) − F (s, vs ))ds. (2.4)
λ→+∞
0
Let g be defined by the right hand side of (2.4). Then, by assumption (H1), we
deduce that there exist positive constants k1 and k2 such that
Z t
|g(t)| ≤ k1 |u0 − v0 | + k2
|uξ − vξ |dξ for t ∈ [0, T ].
0
Using estimate (2.3), we obtain that
Z
|ut − vt | ≤ ke1 |u0 − v0 | + ke2
t
|uξ − vξ |dξ
for t ∈ [0, T ],
0
for some positive constants ke1 and ke2 . Using Gronwall’s Lemma, one obtains the
estimate (2.2).
Consequently, we have the local boundedness of the solutions.
Corollary 2.6. Assume that (H0) and (H1) hold. Then, the solutions of (1.1)
are locally bounded, in the sense that for each B0 > 0 and T0 > 0, there exists a
constant B 0 > 0, such that |ϕ| ≤ B0 implies that |u(t, ϕ)| ≤ B 0 for t ∈ [0, T0 ].
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To study the qualitative behavior of solutions, we need to make additional assumptions on the following difference equation
d
D(wt ) = 0 for t ≥ 0
(2.5)
dt
w0 = ϕ.
The following definition was given for neutral functional differential equation in
finite dimensional spaces, for more details we refer to [17].
Definition 2.7. [5] The operator D is stable if there exist positive constants β
and γ such that the solution of the homogeneous difference equation (2.5) with
w0 = ϕ ∈ {ψ ∈ C; Dψ = 0}, satisfies the following estimate
|wt (., ϕ)| ≤ γ exp(−βt)|ϕ|
for t ≥ 0.
Example 2.8. The operator D defined by
Dϕ = ϕ(0) − qϕ(−r)
is stable if and only if |q| < 1.
Theorem 2.9 ([5, Lemma 2.9]). If the operator D is stable. Then, there are
positive constants a, b, c and d such that for any continuous function h : R+ → X,
the solution w of the difference equation
D(wt ) = h(t) for t ≥ 0
w0 = ϕ ∈ C,
satisfies the estimate
|wt (., ϕ)| ≤ e−at b|ϕ| + c sup |h(s)| + d
s∈[0,t]
sup
|h(s)|
for t ≥ 0.
s∈[max{0,t−r},t]
The Kuratowski’s measure of noncompactness. of bounded sets K on a Banach
space Y is defined by
α(K) = inf{ > 0 : K has a finite cover of ball of diameter less than }.
Lemma 2.10. [21] Let A1 and A2 be bounded sets of a Banach space Y . Then
(i) α(A1 ) ≤ dia(A1 ), where dia(A1 ) = supx,y∈A1 |x − y|,
(ii) α(A1 ) = 0 if and only if A1 is relatively compact in Y ,
(iii) α(A1 ∪ A2 ) = max{α(A1 ), α(A2 )}.
Let K : Y → Y be a closed linear operator with a dense domain D(K) in a
Banach space Y . We denote by σ(K) the spectrum of K.
Definition 2.11 ([25]). The essential spectrum σess (K) of K is the set of all λ ∈ C
such that at least one of the following holds:
(i) The range Im(λI − K) is not closed,
(ii) the generalized eigenspace Mλ (K) = ∪n≥1 ker(λI − K)n of λ is infinite
dimensional,
(iii) λ is a limit point of σ(K), that is λ ∈ σ(K)/{λ}.
For a bounded linear operator K on Y , the Kuratowski measure of non-compactness of K is defined by
|K|α = inf{ > 0 : α(K(B)) ≤ α(B) for every bounded subset B of Y }.
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The essential radius ress (K) is given by
ress (K) = sup{|λ| : λ ∈ σess (K)}.
The computation of essential radius is given by the following Nussbaum’s formula.
Lemma 2.12 ([24]).
ress (K) = lim (|Kn |α )1/n .
n→+∞
Definition 2.13 ([17]). A continuous mapping P : Y → Y is said to be an αcontraction if P maps bounded sets into bounded sets and if there exists a constant
k ∈ (0, 1) such that
α(P (B)) ≤ kα(B),
for every bounded subset B of Y .
Definition 2.14 ([17]). A continuous mapping P : Y → Y is a condensing map
on Y if P maps bounded sets into bounded sets and
α(P (B)) < α(B),
for every bounded subset B of Y such that α(B) > 0.
Let C0 be the phase space of Equation (1.1) defined by
C0 = {ϕ ∈ C : Dϕ ∈ D(A)}.
For each t ≥ 0, we define the linear operator U(t) on C0 by
U(t)ϕ = xt (., ϕ),
where x(., ϕ) is the solution of the equation
d
D(ut ) = AD(ut ) for t ≥ 0
dt
u0 = ϕ ∈ C.
(2.6)
Without loss of generality, we assume that
(H2) (T0 (t))t≥0 is exponentially stable, which means that there exist α0 > 0 and
M0 ≥ 1 such that
|T0 (t)| ≤ M0 e−α0 t
for t ≥ 0.
Otherwise, we can replace A by A − δI, where δ > 0 can be chosen such that the
semigroup generated by the part of A − δI on D(A) is exponentially stable.
We assume that
(H3) D is stable.
The following fundamental lemma plays a crucial role for the existence of periodic
solutions.
Lemma 2.15 ([5, Proposition 2.11]). Assume that (H0), (H2) and (H3) hold.
Then, (U(t))t≥0 is an exponentially stable semigroup on C0 , that is there exist
η > 0 and M ≥ 1 such that
|U(t)| ≤ M e−ηt
for t ≥ 0.
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7
For ϕ ∈ C0 , we introduce the new norm on C0 by
|ϕ|η = sup eηt |U(t)ϕ|,
t≥0
where η is the positive constant given in Lemma 2.15. Clearly,
|ϕ| ≤ |ϕ|η ≤ M |ϕ|,
which implies that |.|η and |.| are equivalent norms on C0 . As an immediate result,
we have the following result.
Corollary 2.16. Assume that (H0), (H2) and (H3) hold. Then
|U(t)|η ≤ e−ηt
for t ≥ 0.
Proof. For every t ≥ 0, one has,
|U(t)ϕ|η = sup eηs |U(s)U(t)ϕ|,
s≥0
= e−ηt sup eη(t+s) |U(s + t)ϕ|,
s≥0
≤ e−ηt sup eηs |U(s)ϕ| = e−ηt |ϕ|η ,
s≥0
which implies |U(t)|η ≤ e−ηt for t ≥ 0.
(H4) T0 (t) is compact on D(A) whenever t > 0.
Theorem 2.17 ([5, Theorem 5.2]). Assume that (H0), (H1), (H2), (H3) and (H4)
hold. Then the solution u(., ϕ) of (1.1) is decomposed as follows:
ut (., ϕ) = U(t)ϕ + W(t)ϕ
for t ≥ 0,
where W(t) is a compact operator on C0 , for each t ≥ 0.
3. Existence of periodic solutions in nonhomogeneous linear case
In this section, we assume that F takes the form
F (t, ϕ) = L(t, ϕ) + f (t)
for t ≥ 0, ϕ ∈ C,
where L is a continuous function from R+ × C into X, linear with respect to the
second argument and f is a continuous function from R into X. Equation (1.1)
becomes
d
D(ut ) = AD(ut ) + L(t, ut ) + f (t) for t ≥ 0,
(3.1)
dt
u0 = ϕ ∈ C,
For the existence of periodic solutions, we assume that
(H5) L and f are ω-periodic in t.
Theorem 3.1. Assume that (H0), (H2), (H3), (H4) and (H5) hold. If Equation
(3.1) has a bounded solution on R+ , then it has an ω-periodic solution.
For the proof, we use Chow and Hale’s fixed point theorem which gives sufficient
conditions for affine maps to have fixed points.
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EJDE-2006/56
Theorem 3.2 ([12]). Let Y be a Banach space and P : Y → Y be an affine map
which is defined by
P x = Sx + y,
where S is a bounded linear operator on Y and y is given in Y . If Im(I − S) is
closed and there exists x0 ∈ Y such that (P n (x0 ))n≥0 is bounded, then P has at
least one fixed point.
Proof of Theorem 3.1. Define the Poincaré map P : C0 → C0 by
ϕ → uω (., ϕ) = uω (., 0, ϕ, L, f ),
where u(., 0, ϕ, L, f ) is the solution of (3.1). By the uniqueness of solutions with
respect to the initial data, ut (., 0, ϕ, L, f ) is decomposed as follows
ut (., 0, ϕ, L, f ) = ut (., 0, ϕ, L, 0) + ut (., 0, 0, L, f )
for t ≥ 0.
Therefore, the Poincaré map P is affine, Pϕ = P0 ϕ+ψ, where P0 ϕ = uω (., 0, ϕ, L, 0)
and ψ = uω (., 0, 0, L, f ). We claim that ress (P0 ) < 1. In fact, by Theorem 2.17, P0
is decomposed as follows
P0 ϕ = U(ω)ϕ + W(ω)ϕ,
where W(ω) is a compact operator on C0 . We deduce that α(P0 ) ≤ α(U(ω)). By
Corollary 2.16, we have
α(P0 ) ≤ exp(−ηω) < 1.
Using Lemma 2.12, we obtain that ress (P0 ) < 1 which implies that 1 is not in the
essential spectrum of P0 . Consequently, Im(I − P0 ) is closed. Let y be the bounded
solution of Equation (3.1) on R+ . Then,
{P n y0 , n ∈ N} = {ynω , n ∈ N},
which gives that (P n y0 )n≥0 is bounded in C0 . By Theorem 3.2, we deduce that P
has at least one fixed point, which gives an ω-periodic solution of (3.1).
4. Boundedness, ultimate boundedness and periodicity
In this section, we study the existence of periodic solutions where the solutions
are bounded and ultimate bounded.
Definition 4.1. The solutions of (1.1) are bounded if for each B1 > 0, there exists
a constant B 1 > 0, such that |ϕ| ≤ B1 implies that |u(t, ϕ)| ≤ B 1 , for t ≥ 0.
Definition 4.2. The solutions of (1.1) are ultimate bounded if there is a bound
B > 0 such that for each B2 > 0, there exists a constant k > 0 such that |ϕ| ≤ B2
and t ≥ k imply that |u(t, ϕ)| ≤ B.
Recall that in [15], the authors have used the concept of boundedness and ultimate boundedeness to prove the existence of a periodic solution for partial functional differential equations of retarded type which correspond to Dϕ = ϕ(0). The
relationship between the local boundedness, the boundedness and the ultimate
boundedeness is given below.
Proposition 4.3. The local boundedness and ultimate boundedness of solutions of
(1.1) imply the boundedness of the solutions.
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9
Proof. Let B be given by the ultimate boundedness, then for any B1 > 0, there
exists a constant k > 0 such that |ϕ| ≤ B1 and t ≥ k imply that |u(t, ϕ)| ≤ B.
Local boundedness of solutions gives that there exists a constant B2 > B such
that |ϕ| ≤ B1 implies that |u(t, ϕ)| < B2 , for t ∈ [0, k]. It follows that for any
positive constant B1 , there exists a constant B2 > B such that |ϕ| ≤ B1 implies
that |u(t, ϕ)| < B2 , for t ≥ 0.
Proposition 4.4. Under assumptions (H0)–(H4), the Poincaré map P is an αcontraction on C0 .
Proof. By Theorem 2.17, P is decomposed as
Pϕ = U(ω)ϕ + W(ω)ϕ,
where W(ω) is a compact operator on C0 . Let Ω a bounded set in C0 . Then
α(P(Ω)) ≤ α(U(ω)(Ω)).
Corollary 2.16 implies
α(P(Ω)) < exp(−ηω)α(Ω)
for any bounded set Ω in C0 ,
which gives that P is an α-contraction map on C0 .
In the following, we assume that
(H6) F is ω-periodic in t.
Theorem 4.5. Assume that (H0), (H1), (H2), (H3), (H4) and (H6) hold. If the
solutions of (1.1) are ultimately bounded, then (1.1) has an ω-periodic solution.
For the proof, we use Hale and Lunel’s fixed point theorem which is an extension
of the well known Horn’s fixed point theorem for condensing maps.
Theorem 4.6 ([17, Hale and Lunel’s fixed point theorem]). Suppose S0 ⊆ S1 ⊆ S2
are convex bounded subsets of a Banach space Y , such that S0 , S2 are closed and
S1 is open in S2 . Let P be a condensing map on Y such that P j (S1 ) ⊆ S2 , for
j ≥ 0, and there is a number N (S1 ) such that P k (S1 ) ⊆ S0 , for k ≥ N (S1 ), then
P has a fixed point.
Proof of Theorem 4.5. By Corollary 2.6 and Proposition 4.3, we know that the
solutions of (1.1) are bounded and ultimate bounded. Let B be the bound in the
definition of ultimate boundedness. By the boundedness of solutions, there exists
a constant B1 > B such that for |ϕ| ≤ B and t ≥ 0, one has |u(t, ϕ)| < B1 .
Moreover, there exists a constant B2 > B1 such that for |ϕ| ≤ B1 and t ≥ 0, then
|u(t, ϕ)| < B2 . By using the ultimate boundedness of solutions of (1.1), we can see
that there exists a positive integer m = m(B1 ) such that for |ϕ| ≤ B1 and t ≥ mω,
we have |u(t, ϕ)| < B. On the other hand,
P j ϕ = ujω (., ϕ)
for j ∈ N.
r
Let k = ω +m+1, where [t] denotes the integer part of t. Then for j ≥ k and |ϕ| ≤
B1 , one has
|P j (ϕ)| = |ujω (., ϕ)| ≤ B,
(4.1)
and for j ∈ {1, 2, . . . , k − 1} and |ϕ| ≤ B1 ,
|P j (ϕ)| = |ujω (., ϕ)| ≤ B2 .
(4.2)
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We define the sets
S0 = {ϕ ∈ C0 : |ϕ| ≤ B},
S1 = {ϕ ∈ C0 : |ϕ| < B1 },
S2 = {ϕ ∈ C0 : |ϕ| ≤ B2 }.
Clearly, S0 , S1 and S2 are convex bounded subsets of the Banach space C0 . Moreover S0 ⊆ S1 ⊆ S2 , S0 and S2 are closed and S1 is open in S2 , and
P j (S1 ) ⊆ S2
for all j ≥ 0.
In fact, inequality (4.1) gives that there exists a positive integer k = k(S1 ) such
that
P j (S1 ) ⊆ S0 ⊆ S2 for j ≥ k,
and from inequality (4.2), we deduce that
P j (S1 ) ⊆ S2
for j ∈ {1, 2, . . . , k − 1}.
By Proposition 4.4, P is an α-contraction map on C0 . Consequently, Theorem 4.6
gives that the Poincaré map P has at least one fixed point which gives an ω-periodic
solution of (1.1).
5. Applications
Nonhomogeneous linear case. To illustrate our previous results, we consider
the following model of continuous circular array of resistively coupled transmission
lines which is taken from [27]
∂
∂2
[u(t, x) − qu(t − r, x)] =
[u(t, x) − qu(t − r, x)] + a1 (t)u(t − r, x)
∂t
∂x2
+ h1 (t, x) for t ≥ 0, x ∈ [0, π],
(5.1)
[u(t, x) − qu(t − r, x)]x=0,π = 0 for t ≥ 0,
u(θ, x) = ϕ0 (θ, x)
for θ ∈ [−r, 0], x ∈ [0, π],
where a1 : R → R, h1 : R × [0, π] → R and ϕ0 : [−r, 0] × [0, π] → R are continuous
functions and 0 < q < 1. Let X = C([0, π]; R) be the space of continuous functions
from [0, π] to R endowed with the uniform norm topology. In order to rewrite (5.1)
in abstract form, we introduce the linear operator A : D(A) ⊂ X → X defined by
D(A) = {y ∈ C 2 ([0, π]; R) : y(0) = y(π) = 0},
Ay = y 00 .
Lemma 5.1 ([13]). (0, +∞) ⊂ ρ(A) and |(λI − A)−1 | ≤
1
λ
for λ > 0.
The above lemma implies that assumption (H0 ) is satisfied. Moreover, one has
D(A) = {y ∈ X : y(0) = y(π) = 0}.
Let D : C → X and L : R × C → X be the bounded linear operators defined
respectively by
Dϕ := ϕ(0) − qϕ(−r),
L(t, ϕ) = a1 (t) ϕ(−r)
for t ∈ R, ϕ ∈ C([−r, 0]; X).
Let f : R −→ X be given by
f (t)(x) = h1 (t, x)
for t ∈ R, x ∈ [0, π].
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11
Then, Equation (5.1) takes the abstract form (3.1). Since 0 < q < 1, the operator
D is stable. The part A0 of A in D(A) is defined by
D(A0 ) = {y ∈ C 2 ([0, π]; R) : y(0) = y(π) = y 00 (0) = y 00 (π) = 0},
A0 y = y 00 .
Lemma 5.2. [10] A0 generates a compact strongly continuous semigroup (T0 (t))t≥0
on D(A) such that
|T0 (t)| ≤ e−t for t ≥ 0.
Therefore, the assumptions (H2) and (H4) hold. Consequently, for any ϕ ∈ C
such that
Dϕ ∈ {y ∈ X : y(0) = y(π) = 0} .
Equation (3.1) has a unique solution u on [−r, +∞). To establish the existence of
a periodic solution of (3.1), we suppose that
(H7) a1 and h1 are ω-periodic in t.
(H8) There exists a positive constant β ∈ (0, 1) such that |a1 |∞ ≤ (1−q)β, where
|a1 |∞ = sups∈R |a1 (s)|.
Proposition 5.3. Assume that (H7) and (H8) hold. Then (3.1) has an ω-periodic
solution.
1
1+
Proof. We will first show that (1.1) has a bounded solution on R+ . Let ρ = 1+q
|f |∞
and ϕ ∈ C such that |ϕ| < ρ. Then |ϕ(0) − qϕ(−r)| < (1 + q)ρ. We claim
1−β
that
|u(t) − qu(t − r)| ≤ (1 + q)ρ for all t ≥ 0.
(5.2)
We proceed by contradiction. Let t0 be the first time such that (5.2) is not true.
Then,
t0 = inf{t > 0 : |u(t) − qu(t − r)| > (1 + q)ρ}.
By continuity, one can see that
|u(t0 ) − qu(t0 − r)| = (1 + q)ρ,
and there exists a positive constant ε > 0 such that
|u(t) − qu(t − r)| > (1 + q)ρ for t ∈ (t0 , t0 + ε).
Using the variation-of-constants formula (2.1), we get that
Z t0
|u(t0 ) − qu(t0 − r)| ≤ e−t0 (1 + q)ρ +
e−(t0 −s) [|a1 |∞ |u(s + θ)|dθ + |f |∞ ]ds.
0
Since |u(t) − qu(t − r)| ≤ (1 + q)ρ for t ≤ t0 , then
|u(t)| ≤ (1 + q)ρ + q|u(t − r)| for t ∈ [−r, t0 ].
|ϕ| < ρ, then we can see that
|u(t)| ≤
1+q
ρ for t ∈ [−r, t0 ],
1−q
and
|u(t0 ) − qu(t0 − r)| ≤ e−t0 (1 + q)ρ + (1 − e−t0 )[
1+q
|a1 |∞ ρ + |f |∞ ].
1−q
Using hypotheses (H8), we obtain
|u(t0 ) − qu(t0 − r)| ≤ e−t0 (1 + q)ρ + (1 − e−t0 )(β(1 + q)ρ + |f |∞ ),
12
R. BENKHALTI, A. ELAZZOUZI, K. EZZINBI
EJDE-2006/56
consequently,
|u(t0 ) − qu(t0 − r)| ≤ (1 + q)ρ − (1 − e−t0 )((1 − β)(1 + q)ρ − |f |∞ ),
|u(t0 ) − qu(t0 − r)| ≤ (1 + q)ρ − (1 − e−t0 )(1 − β) < (1 + q)ρ.
By continuity, there exists a positive ε0 such that
|u(t) − qu(t − r)| < (1 + q)ρ
for t ∈ (t0 , t0 + ε0 ),
which gives a contradiction and we deduce that
|u(t) − qu(t − r)| ≤ (1 + q)ρ
for t ≥ 0.
Let t ∈ [0, r]. Then
|u(t)| ≤ (1 + q)ρ + qρ ≤ (1 + q)(1 + q)ρ,
and for t ∈ [r, 2r],
|u(t)| ≤ (1 + q)(1 + q + q 2 )ρ.
We proceed by steps, then for t ∈ [(n − 1)r, nr], we have
|u(t)| ≤ (1 + q)(1 + q + q 2 + · · · + q n )ρ.
Consequently,
|u(t)| ≤ (1 + q)ρ
X
qn =
n≥0
1+q
ρ for all t ≥ 0.
1−q
Then, Equation (3.1) has a bounded solution u on R+ . By Theorem 3.1, we deduce
that Equation (3.1) has an ω-periodic solution.
Nonlinear case. We consider the nonlinear equation
∂2
∂
[u(t, x) − qu(t − r, x)] =
[u(t, x) − qu(t − r, x)] + a2 (t)g1 (u(t − r, x))
∂t
∂x2
+ h2 (t, x) for t ≥ 0, x ∈ [0, π],
(5.3)
[u(t, x) − qu(t − r, x)]x=0,π = 0 for t ≥ 0,
u(θ, x) = ϕ0 (θ, x)
for θ ∈ [−r, 0], x ∈ [0, π],
where g1 : R → R is a Lipschitz continuous function and a2 , ϕ0 : [−r, 0]×[0, π] → R
are continuous functions and 0 < q < 1. We define the function F : R × C → X by
F (t, ϕ)(x) = a2 (t)g1 (ϕ(−r)(x)) + h2 (t, x)
for t ∈ R, x ∈ [0, π], ϕ ∈ C.
Then, (5.3) takes the abstract form (1.1).
We assume that
(H9) a2 , h2 are ω-periodic in t.
(H10) g1 is bounded on R.
Proposition 5.4. Assume that (H9) and (H10) hold. Then, the solutions of (1.1)
are ultimately bounded.
Proof. Since 0 < q < 1, then the operator D is stable. By Theorem 2.5, we deduce
that there exist positive constants a, b and c such that
|ut (., ϕ)| ≤ ae−bt |ϕ| + sup |h(s)| + c
sup
|h(s)|,
(5.4)
s∈[0,t]
s∈[max{0,t−r},t]
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PERIODIC SOLUTIONS
where
Z
t
T0 (t − s)Bλ F (s, us )ds for t ≥ 0.
h(t) = T0 (t)Dϕ + lim
λ→+∞
13
0
Using Assumption (H10) and the fact that |T0 (t)| ≤ e−t
there exist positive constants e
a and eb, such that
for t ≥ 0, we obtain that
|h(t)| ≤ e
ae−t |ϕ| + eb for t ≥ 0,
which implies for t > r that
sup |h(s)| ≤ e
aer−t |ϕ| + eb for ϕ ∈ C.
s∈[t−r,t]
Using the estimate (5.4), we obtain
|ut (., ϕ)| ≤ ae−bt |ϕ| + c for t > r ϕ ∈ C,
for some positive constants a, b and c. Consequently, there exists a positive constant
e such that
K
e for ϕ ∈ C,
lim sup |u(t, ϕ)| < K
t→+∞
and we deduce that the solutions of (1.1) are ultimately bounded.
Consequently by Theorem 4.5, we obtain the following result.
Proposition 5.5. Assume that (H9) and (H10) hold. Then (1.1) has an ω-periodic
solution.
Acknowledgments. The authors would like to thank the anonymous referee for
his/her careful reading of the original version.
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Rachid Benkhalti
Pacific Lutheran University, Department of Mathematics, Tacoma, Washington, 98447,
USA
E-mail address: benkhar@plu.edu
Abdelhai Elazzouzi
Université Cadi Ayyad, Faculté des Sciences Semlalia, Département de Mathématiques,
B.P. 2390 Marrakesh, Morocco
E-mail address: a.elazzouzi@ucam.ac.ma
Khalil Ezzinbi
Université Cadi Ayyad, Faculté des Sciences Semlalia, Département de Mathématiques,
B.P. 2390 Marrakesh, Morocco
E-mail address: kezzinbi@ictp.it, ezzinbi@ucam.ac.ma